Computational Aspects of the Combinatorial Nullstellensatz Method

Combinatorial Nullstellensatz Modulo Prime Powers and the Parity Argument

We present new generalizations of Olson’s theorem and of a consequence of Alon’s Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degreeconstrain...

Combinatorial Nullstellensatz

The Combinatorial Nullstellensatz is a theorem about the roots of a polynomial. It is related to Hilbert’s Nullstellensatz. Established in 1996 by Alon et al. [4] and generalized in 1999 by Alon [2], the Combinatorial Nullstellensatz is a powerful tool that allows the use of polynomials to solve problems in number theory and graph theory. This article introduces the Combinatorial Nullstellensat...

A Generalization of Combinatorial Nullstellensatz

In this note we give an extended version of Combinatorial Nullstellensatz, with weaker assumption on nonvanishing monomial. We also present an application of our result in a situation where the original theorem does not seem to work.

No feasible monotone interpolation for simple combinatorial reasoning

The feasible monotone interpolation method has been one of the main tools to prove the exponential lower bounds for relatively weak propositional systems. In [1], we introduced a simple combinatorial reasoning system, GCNF+permutation, as a candidate for an automatizable, though powerful, propositional calculus. We show that the monotone interpolation method is not applicable to prove the super...

Applications of the Combinatorial Nullstellensatz